For half a century, geostatistical methods have been increasingly used in the petroleum industry for modeling geological and petrophysical heterogeneities of hydrocarbon reservoirs. Traditional geostatistical methods are based on random function models that are defined to mimic the internal geological architecture of the hydrocarbon reservoir of interest. However, in some cases, for example a meandering channel system, random function based methods are either not suitable to capture the complexity of the geological patterns/features, or not flexible enough for data conditioning.
In the last two decades, multiple-point (MP) geostatistics has been developed and increasingly used for modeling subsurface heterogeneity (Guardiano and Srivastava, 1993; Strebelle, 2000; Hu and Chugunova, 2008). Unlike traditional geostatistical simulations based on random function models, a MP geostatistical simulation does not require the explicit definition of a random function. Instead, it directly utilizes empirical multivariate distributions inferred from one or more training images (TI's). This approach is flexible to data conditioning and to representing complex architectures of geological facies and petrophysical properties.
In general, training images should represent the spatial distribution of the geometrical patterns/features of the reservoir heterogeneity. However, current MPS algorithms require the training images to be spatially stationary. This means that the training image, being stationary, bears no information about the location of the geometrical patterns/features of heterogeneity in either the reservoir itself or in a model realization. Instead, the stationary TI provides a population of cells having certain characteristics, such as a meandering channel system with channel width, thickness and degree of meandering. The statistical distribution of the characteristics is intended to reflect the statistical distribution of those characteristics in the reservoir which is being modeled, but without reference to location. The TI need not have the same dimensions as the model and often has fewer cells in the vertical axis but more cells in the horizontal axes than the model. The stationary TI is actually far from being a reservoir model.
However, real geological patterns often present spatial trends and are thus not stationary in the sense described above. Normally, a geologist or a geomodeller will need to create a TI prior to a model being created. Creating a realistic, but stationary TI is a contradictory task for geologists and geomodellers, simply because a realistic TI cannot be stationary in most real situations.
Methods have been developed to integrate spatial trends into MPS realizations (see, e.g. Strebelle and Zhang, 2005), but these method still use stationary TI's.
Chugunova and Hu (2008) describe a MPS method with non-stationary training images. In their method, coupled primary and secondary training images are used to infer the conditional probability of the primary variable given a primary pattern and a secondary datum. This method can be applied to the case where a secondary data set (e.g., from seismic) is available for constraining the spatial distribution of geological patterns. Although realistic MPS models are constructed by using this method, the basic algorithm remains heuristic. It also requires building a secondary training image from the primary training image in consistency with the secondary data. Besides, the non-stationary TI's of the above MPS method do not necessarily reflect the location of the geometrical patterns/features of the reservoir heterogeneity. Therefore, they are also far from being a reservoir model.